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Theorem ersym 6029
Description: An equivalence relation is symmetric. (Contributed by NM, 4-Jun-1995.) (Revised by Mario Carneiro, 12-Aug-2015.)
Hypotheses
Ref Expression
ersym.1 (φ𝑅 Er 𝑋)
ersym.2 (φA𝑅B)
Assertion
Ref Expression
ersym (φB𝑅A)

Proof of Theorem ersym
StepHypRef Expression
1 ersym.2 . . 3 (φA𝑅B)
2 ersym.1 . . . . . 6 (φ𝑅 Er 𝑋)
3 errel 6026 . . . . . 6 (𝑅 Er 𝑋 → Rel 𝑅)
42, 3syl 14 . . . . 5 (φ → Rel 𝑅)
5 brrelex12 4308 . . . . 5 ((Rel 𝑅 A𝑅B) → (A V B V))
64, 1, 5syl2anc 393 . . . 4 (φ → (A V B V))
7 brcnvg 4443 . . . . 5 ((B V A V) → (B𝑅AA𝑅B))
87ancoms 255 . . . 4 ((A V B V) → (B𝑅AA𝑅B))
96, 8syl 14 . . 3 (φ → (B𝑅AA𝑅B))
101, 9mpbird 156 . 2 (φB𝑅A)
11 df-er 6017 . . . . . 6 (𝑅 Er 𝑋 ↔ (Rel 𝑅 dom 𝑅 = 𝑋 (𝑅 ∪ (𝑅𝑅)) ⊆ 𝑅))
1211simp3bi 909 . . . . 5 (𝑅 Er 𝑋 → (𝑅 ∪ (𝑅𝑅)) ⊆ 𝑅)
132, 12syl 14 . . . 4 (φ → (𝑅 ∪ (𝑅𝑅)) ⊆ 𝑅)
1413unssad 3097 . . 3 (φ𝑅𝑅)
1514ssbrd 3779 . 2 (φ → (B𝑅AB𝑅A))
1610, 15mpd 13 1 (φB𝑅A)
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98   = wceq 1228   wcel 1374  Vcvv 2535  cun 2892  wss 2894   class class class wbr 3738  ccnv 4271  dom cdm 4272  ccom 4276  Rel wrel 4277   Er wer 6014
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-14 1386  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-sep 3849  ax-pow 3901  ax-pr 3918
This theorem depends on definitions:  df-bi 110  df-3an 875  df-tru 1231  df-nf 1330  df-sb 1628  df-eu 1885  df-mo 1886  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ral 2289  df-rex 2290  df-v 2537  df-un 2899  df-in 2901  df-ss 2908  df-pw 3336  df-sn 3356  df-pr 3357  df-op 3359  df-br 3739  df-opab 3793  df-xp 4278  df-rel 4279  df-cnv 4280  df-er 6017
This theorem is referenced by:  ercl2  6030  ersymb  6031  ertr2d  6034  ertr3d  6035  ertr4d  6036  erth  6061  erinxp  6091
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